# Preasymptotic estimates for approximation of multivariate periodic Sobolev functions

ASCW01 - Challenges in optimal recovery and hyperbolic cross approximation

Approximation of Sobolev functions is a topic with a long history and many applications in different branches of mathematics. The asymptotic order as $n\to\infty$ of the approximation numbers $a_n$ is well-known for embeddings of isotropic Sobolev spaces and also for Sobolev spaces of dominating mixed smoothness. However, if the dimension $d$ of the underlying domain is very high, one has to wait exponentially long until the asymptotic rate becomes visible. Hence, for computational issues this rate is useless, what really matters is the preasymptotic range, say $n\le 2d$.
In the talk I will first give a short overview over this relatively new field. Then I will present some new preasymptotic estimates for $L_2$-approximation of periodic Sobolev functions, which improve the previously known results. I will discuss the cases of isotropic and dominating mixed smoothness, and also $C \infty$-functions of Gevrey type. Clearly, on all these spaces there are many equivalent norms. It is an interesting effect that – in contrast to the asymptotic rates – the preasymptotic behaviour strongly depends on the chosen norm.

This talk is part of the Isaac Newton Institute Seminar Series series.